TSTP Solution File: GRP682+1 by Twee---2.4.2
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- Process Solution
%------------------------------------------------------------------------------
% File : Twee---2.4.2
% Problem : GRP682+1 : TPTP v8.1.2. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% Computer : n032.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8042.1875MB
% OS : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Thu Aug 31 01:19:41 EDT 2023
% Result : Theorem 0.13s 0.43s
% Output : Proof 0.13s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.05/0.09 % Problem : GRP682+1 : TPTP v8.1.2. Released v4.0.0.
% 0.05/0.09 % Command : parallel-twee %s --tstp --conditional-encoding if --smaller --drop-non-horn --give-up-on-saturation --explain-encoding --formal-proof
% 0.09/0.28 % Computer : n032.cluster.edu
% 0.09/0.28 % Model : x86_64 x86_64
% 0.09/0.28 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.09/0.28 % Memory : 8042.1875MB
% 0.09/0.28 % OS : Linux 3.10.0-693.el7.x86_64
% 0.09/0.28 % CPULimit : 300
% 0.09/0.28 % WCLimit : 300
% 0.09/0.28 % DateTime : Tue Aug 29 00:39:46 EDT 2023
% 0.09/0.28 % CPUTime :
% 0.13/0.43 Command-line arguments: --no-flatten-goal
% 0.13/0.43
% 0.13/0.43 % SZS status Theorem
% 0.13/0.43
% 0.13/0.44 % SZS output start Proof
% 0.13/0.44 Take the following subset of the input axioms:
% 0.13/0.44 fof(f01, axiom, ![A]: ld(A, mult(A, A))=A).
% 0.13/0.44 fof(f02, axiom, ![A2]: rd(mult(A2, A2), A2)=A2).
% 0.13/0.44 fof(f03, axiom, ![B, A2]: mult(A2, ld(A2, B))=ld(A2, mult(A2, B))).
% 0.13/0.44 fof(f04, axiom, ![A2, B2]: mult(rd(A2, B2), B2)=rd(mult(A2, B2), B2)).
% 0.13/0.44 fof(f05, axiom, ![D, C, A2, B2]: ld(ld(A2, B2), mult(ld(A2, B2), mult(C, D)))=mult(ld(A2, mult(A2, C)), D)).
% 0.13/0.44 fof(f06, axiom, ![A2, B2, D2, C2]: rd(mult(mult(A2, B2), rd(C2, D2)), rd(C2, D2))=mult(A2, rd(mult(B2, D2), D2))).
% 0.13/0.44 fof(f07, axiom, ![A2, B2]: ld(A2, mult(A2, ld(B2, B2)))=rd(mult(rd(A2, A2), B2), B2)).
% 0.13/0.44 fof(goals, conjecture, ![X0, X1, X2]: (ld(ld(X0, X1), mult(ld(X0, X1), X2))=ld(X0, mult(X0, X2)) & ld(rd(X0, X1), mult(rd(X0, X1), X2))=ld(X0, mult(X0, X2)))).
% 0.13/0.44
% 0.13/0.44 Now clausify the problem and encode Horn clauses using encoding 3 of
% 0.13/0.44 http://www.cse.chalmers.se/~nicsma/papers/horn.pdf.
% 0.13/0.44 We repeatedly replace C & s=t => u=v by the two clauses:
% 0.13/0.44 fresh(y, y, x1...xn) = u
% 0.13/0.44 C => fresh(s, t, x1...xn) = v
% 0.13/0.44 where fresh is a fresh function symbol and x1..xn are the free
% 0.13/0.44 variables of u and v.
% 0.13/0.44 A predicate p(X) is encoded as p(X)=true (this is sound, because the
% 0.13/0.44 input problem has no model of domain size 1).
% 0.13/0.44
% 0.13/0.44 The encoding turns the above axioms into the following unit equations and goals:
% 0.13/0.44
% 0.13/0.44 Axiom 1 (f02): rd(mult(X, X), X) = X.
% 0.13/0.44 Axiom 2 (f01): ld(X, mult(X, X)) = X.
% 0.13/0.44 Axiom 3 (f03): mult(X, ld(X, Y)) = ld(X, mult(X, Y)).
% 0.13/0.44 Axiom 4 (f04): mult(rd(X, Y), Y) = rd(mult(X, Y), Y).
% 0.13/0.44 Axiom 5 (f07): ld(X, mult(X, ld(Y, Y))) = rd(mult(rd(X, X), Y), Y).
% 0.13/0.44 Axiom 6 (f06): rd(mult(mult(X, Y), rd(Z, W)), rd(Z, W)) = mult(X, rd(mult(Y, W), W)).
% 0.13/0.44 Axiom 7 (f05): ld(ld(X, Y), mult(ld(X, Y), mult(Z, W))) = mult(ld(X, mult(X, Z)), W).
% 0.13/0.44
% 0.13/0.44 Lemma 8: ld(X, X) = rd(X, X).
% 0.13/0.44 Proof:
% 0.13/0.44 ld(X, X)
% 0.13/0.44 = { by axiom 2 (f01) R->L }
% 0.13/0.44 ld(X, ld(X, mult(X, X)))
% 0.13/0.44 = { by axiom 3 (f03) R->L }
% 0.13/0.44 ld(X, mult(X, ld(X, X)))
% 0.13/0.44 = { by axiom 5 (f07) }
% 0.13/0.44 rd(mult(rd(X, X), X), X)
% 0.13/0.44 = { by axiom 4 (f04) }
% 0.13/0.44 rd(rd(mult(X, X), X), X)
% 0.13/0.44 = { by axiom 1 (f02) }
% 0.13/0.44 rd(X, X)
% 0.13/0.44
% 0.13/0.44 Lemma 9: mult(X, rd(X, X)) = X.
% 0.13/0.44 Proof:
% 0.13/0.44 mult(X, rd(X, X))
% 0.13/0.44 = { by lemma 8 R->L }
% 0.13/0.44 mult(X, ld(X, X))
% 0.13/0.44 = { by axiom 3 (f03) }
% 0.13/0.44 ld(X, mult(X, X))
% 0.13/0.44 = { by axiom 2 (f01) }
% 0.13/0.44 X
% 0.13/0.44
% 0.13/0.44 Lemma 10: mult(ld(X, mult(X, Y)), Z) = ld(X, mult(X, mult(Y, Z))).
% 0.13/0.44 Proof:
% 0.13/0.44 mult(ld(X, mult(X, Y)), Z)
% 0.13/0.44 = { by axiom 7 (f05) R->L }
% 0.13/0.44 ld(ld(X, mult(X, X)), mult(ld(X, mult(X, X)), mult(Y, Z)))
% 0.13/0.44 = { by axiom 2 (f01) }
% 0.13/0.44 ld(X, mult(ld(X, mult(X, X)), mult(Y, Z)))
% 0.13/0.44 = { by axiom 2 (f01) }
% 0.13/0.44 ld(X, mult(X, mult(Y, Z)))
% 0.13/0.44
% 0.13/0.44 Lemma 11: ld(X, mult(X, mult(X, Y))) = mult(X, Y).
% 0.13/0.44 Proof:
% 0.13/0.44 ld(X, mult(X, mult(X, Y)))
% 0.13/0.44 = { by lemma 10 R->L }
% 0.13/0.44 mult(ld(X, mult(X, X)), Y)
% 0.13/0.44 = { by axiom 2 (f01) }
% 0.13/0.44 mult(X, Y)
% 0.13/0.44
% 0.13/0.44 Lemma 12: mult(X, rd(mult(Y, Z), Z)) = rd(mult(mult(X, Y), Z), Z).
% 0.13/0.44 Proof:
% 0.13/0.44 mult(X, rd(mult(Y, Z), Z))
% 0.13/0.44 = { by axiom 6 (f06) R->L }
% 0.13/0.44 rd(mult(mult(X, Y), rd(mult(Z, Z), Z)), rd(mult(Z, Z), Z))
% 0.13/0.44 = { by axiom 1 (f02) }
% 0.13/0.44 rd(mult(mult(X, Y), Z), rd(mult(Z, Z), Z))
% 0.13/0.44 = { by axiom 1 (f02) }
% 0.13/0.44 rd(mult(mult(X, Y), Z), Z)
% 0.13/0.44
% 0.13/0.44 Lemma 13: mult(X, rd(Y, Y)) = rd(mult(X, Y), Y).
% 0.13/0.44 Proof:
% 0.13/0.44 mult(X, rd(Y, Y))
% 0.13/0.44 = { by lemma 11 R->L }
% 0.13/0.44 ld(X, mult(X, mult(X, rd(Y, Y))))
% 0.13/0.44 = { by axiom 3 (f03) R->L }
% 0.13/0.44 mult(X, ld(X, mult(X, rd(Y, Y))))
% 0.13/0.44 = { by lemma 8 R->L }
% 0.13/0.44 mult(X, ld(X, mult(X, ld(Y, Y))))
% 0.13/0.44 = { by axiom 5 (f07) }
% 0.13/0.44 mult(X, rd(mult(rd(X, X), Y), Y))
% 0.13/0.44 = { by lemma 12 }
% 0.13/0.44 rd(mult(mult(X, rd(X, X)), Y), Y)
% 0.13/0.44 = { by lemma 9 }
% 0.13/0.44 rd(mult(X, Y), Y)
% 0.13/0.44
% 0.13/0.44 Lemma 14: rd(mult(X, rd(Y, Z)), rd(Y, Z)) = rd(mult(X, Z), Z).
% 0.13/0.44 Proof:
% 0.13/0.44 rd(mult(X, rd(Y, Z)), rd(Y, Z))
% 0.13/0.44 = { by lemma 9 R->L }
% 0.13/0.44 rd(mult(mult(X, rd(X, X)), rd(Y, Z)), rd(Y, Z))
% 0.13/0.44 = { by axiom 6 (f06) }
% 0.13/0.44 mult(X, rd(mult(rd(X, X), Z), Z))
% 0.13/0.44 = { by lemma 12 }
% 0.13/0.44 rd(mult(mult(X, rd(X, X)), Z), Z)
% 0.13/0.44 = { by lemma 9 }
% 0.13/0.44 rd(mult(X, Z), Z)
% 0.13/0.44
% 0.13/0.44 Lemma 15: rd(mult(mult(X, Y), Y), Y) = mult(X, Y).
% 0.13/0.44 Proof:
% 0.13/0.44 rd(mult(mult(X, Y), Y), Y)
% 0.13/0.44 = { by lemma 12 R->L }
% 0.13/0.44 mult(X, rd(mult(Y, Y), Y))
% 0.13/0.44 = { by axiom 1 (f02) }
% 0.13/0.44 mult(X, Y)
% 0.13/0.44
% 0.13/0.44 Lemma 16: rd(mult(X, Y), Y) = rd(X, rd(Y, Y)).
% 0.13/0.44 Proof:
% 0.13/0.44 rd(mult(X, Y), Y)
% 0.13/0.44 = { by lemma 14 R->L }
% 0.13/0.44 rd(mult(X, rd(Y, Y)), rd(Y, Y))
% 0.13/0.45 = { by axiom 4 (f04) R->L }
% 0.13/0.45 mult(rd(X, rd(Y, Y)), rd(Y, Y))
% 0.13/0.45 = { by lemma 9 R->L }
% 0.13/0.45 mult(rd(X, rd(Y, Y)), mult(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))))
% 0.13/0.45 = { by lemma 15 R->L }
% 0.13/0.45 mult(rd(X, rd(Y, Y)), rd(mult(mult(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(rd(Y, Y), rd(Y, Y))), rd(rd(Y, Y), rd(Y, Y))))
% 0.13/0.45 = { by lemma 9 }
% 0.13/0.45 mult(rd(X, rd(Y, Y)), rd(mult(rd(Y, Y), rd(rd(Y, Y), rd(Y, Y))), rd(rd(Y, Y), rd(Y, Y))))
% 0.13/0.45 = { by lemma 12 }
% 0.13/0.45 rd(mult(mult(rd(X, rd(Y, Y)), rd(Y, Y)), rd(rd(Y, Y), rd(Y, Y))), rd(rd(Y, Y), rd(Y, Y)))
% 0.13/0.45 = { by lemma 13 }
% 0.13/0.45 rd(rd(mult(mult(rd(X, rd(Y, Y)), rd(Y, Y)), rd(Y, Y)), rd(Y, Y)), rd(rd(Y, Y), rd(Y, Y)))
% 0.13/0.45 = { by lemma 15 }
% 0.13/0.45 rd(mult(rd(X, rd(Y, Y)), rd(Y, Y)), rd(rd(Y, Y), rd(Y, Y)))
% 0.13/0.45 = { by axiom 4 (f04) }
% 0.13/0.45 rd(rd(mult(X, rd(Y, Y)), rd(Y, Y)), rd(rd(Y, Y), rd(Y, Y)))
% 0.13/0.45 = { by axiom 1 (f02) R->L }
% 0.13/0.45 rd(rd(mult(X, rd(Y, Y)), rd(Y, Y)), rd(rd(rd(mult(Y, Y), Y), Y), rd(Y, Y)))
% 0.13/0.45 = { by axiom 4 (f04) R->L }
% 0.13/0.45 rd(rd(mult(X, rd(Y, Y)), rd(Y, Y)), rd(rd(mult(rd(Y, Y), Y), Y), rd(Y, Y)))
% 0.13/0.45 = { by lemma 13 R->L }
% 0.13/0.45 rd(rd(mult(X, rd(Y, Y)), rd(Y, Y)), rd(mult(rd(Y, Y), rd(Y, Y)), rd(Y, Y)))
% 0.13/0.45 = { by axiom 1 (f02) }
% 0.13/0.45 rd(rd(mult(X, rd(Y, Y)), rd(Y, Y)), rd(Y, Y))
% 0.13/0.45 = { by axiom 4 (f04) R->L }
% 0.13/0.45 rd(mult(rd(X, rd(Y, Y)), rd(Y, Y)), rd(Y, Y))
% 0.13/0.45 = { by lemma 14 R->L }
% 0.13/0.45 rd(mult(rd(X, rd(Y, Y)), rd(X, rd(Y, Y))), rd(X, rd(Y, Y)))
% 0.13/0.45 = { by axiom 1 (f02) }
% 0.13/0.45 rd(X, rd(Y, Y))
% 0.13/0.45
% 0.13/0.45 Lemma 17: ld(ld(X, Y), mult(ld(X, Y), Z)) = ld(X, mult(X, Z)).
% 0.13/0.45 Proof:
% 0.13/0.45 ld(ld(X, Y), mult(ld(X, Y), Z))
% 0.13/0.45 = { by lemma 9 R->L }
% 0.13/0.45 ld(ld(X, Y), mult(ld(X, Y), mult(Z, rd(Z, Z))))
% 0.13/0.45 = { by axiom 7 (f05) }
% 0.13/0.45 mult(ld(X, mult(X, Z)), rd(Z, Z))
% 0.13/0.45 = { by lemma 10 }
% 0.13/0.45 ld(X, mult(X, mult(Z, rd(Z, Z))))
% 0.13/0.45 = { by lemma 9 }
% 0.13/0.45 ld(X, mult(X, Z))
% 0.13/0.45
% 0.13/0.45 Lemma 18: ld(mult(X, Y), mult(mult(X, Y), Z)) = ld(X, mult(X, Z)).
% 0.13/0.45 Proof:
% 0.13/0.45 ld(mult(X, Y), mult(mult(X, Y), Z))
% 0.13/0.45 = { by lemma 11 R->L }
% 0.13/0.45 ld(mult(X, Y), mult(ld(X, mult(X, mult(X, Y))), Z))
% 0.13/0.45 = { by lemma 11 R->L }
% 0.13/0.45 ld(ld(X, mult(X, mult(X, Y))), mult(ld(X, mult(X, mult(X, Y))), Z))
% 0.13/0.45 = { by lemma 17 }
% 0.13/0.45 ld(X, mult(X, Z))
% 0.13/0.45
% 0.13/0.45 Goal 1 (goals): tuple(ld(ld(x0_2, x1_2), mult(ld(x0_2, x1_2), x2_2)), ld(rd(x0, x1), mult(rd(x0, x1), x2))) = tuple(ld(x0_2, mult(x0_2, x2_2)), ld(x0, mult(x0, x2))).
% 0.13/0.45 Proof:
% 0.13/0.45 tuple(ld(ld(x0_2, x1_2), mult(ld(x0_2, x1_2), x2_2)), ld(rd(x0, x1), mult(rd(x0, x1), x2)))
% 0.13/0.45 = { by lemma 17 }
% 0.13/0.45 tuple(ld(x0_2, mult(x0_2, x2_2)), ld(rd(x0, x1), mult(rd(x0, x1), x2)))
% 0.13/0.45 = { by lemma 18 R->L }
% 0.13/0.45 tuple(ld(x0_2, mult(x0_2, x2_2)), ld(mult(rd(x0, x1), x1), mult(mult(rd(x0, x1), x1), x2)))
% 0.13/0.45 = { by axiom 4 (f04) }
% 0.13/0.45 tuple(ld(x0_2, mult(x0_2, x2_2)), ld(rd(mult(x0, x1), x1), mult(mult(rd(x0, x1), x1), x2)))
% 0.13/0.45 = { by lemma 16 }
% 0.13/0.45 tuple(ld(x0_2, mult(x0_2, x2_2)), ld(rd(x0, rd(x1, x1)), mult(mult(rd(x0, x1), x1), x2)))
% 0.13/0.45 = { by axiom 4 (f04) }
% 0.13/0.45 tuple(ld(x0_2, mult(x0_2, x2_2)), ld(rd(x0, rd(x1, x1)), mult(rd(mult(x0, x1), x1), x2)))
% 0.13/0.45 = { by lemma 13 R->L }
% 0.13/0.45 tuple(ld(x0_2, mult(x0_2, x2_2)), ld(rd(x0, rd(x1, x1)), mult(mult(x0, rd(x1, x1)), x2)))
% 0.13/0.45 = { by lemma 16 R->L }
% 0.13/0.45 tuple(ld(x0_2, mult(x0_2, x2_2)), ld(rd(mult(x0, x1), x1), mult(mult(x0, rd(x1, x1)), x2)))
% 0.13/0.45 = { by lemma 13 R->L }
% 0.13/0.45 tuple(ld(x0_2, mult(x0_2, x2_2)), ld(mult(x0, rd(x1, x1)), mult(mult(x0, rd(x1, x1)), x2)))
% 0.13/0.45 = { by lemma 18 }
% 0.13/0.45 tuple(ld(x0_2, mult(x0_2, x2_2)), ld(x0, mult(x0, x2)))
% 0.13/0.45 % SZS output end Proof
% 0.13/0.45
% 0.13/0.45 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------